The Review of Regional Studies, Vol. 37, No. 1, 2007, pp. 28 – 38
ISSN 1553-0892
Spatial Structure and Spatial Interaction: 25 Years Later Daniel A. Griffith School of Economic, Political and Policy Sciences, University of Texas at Dallas, Richardson, TX 75083-0688, e-mail:
[email protected]
Abstract In the 1970s, spatial autocorrelation (i.e., local distance and configuration effects) and distance decay (i.e., global distance effects) were suspected of being intermingled in spatial interaction model specifications. This convolution was first treated in a theoretical context by Curry (1972), with some subsequent debate (e.g., Curry, Griffith, and Sheppard 1975). This work was followed by a documentation of the convolution (e.g., Griffith and Jones 1980) and further theoretical treatment of the role spatial autocorrelation plays in spatial interaction modeling (e.g., Griffith 1982). But methodology did not exist at the time—or even soon thereafter—to easily or fully address spatial autocorrelation effects within spatial interaction model specifications, a contention attested to and demonstrated by the cumbersome and difficult-to-implement techniques employed by, for example, Bolduc, Laferrière, and Santarossa (1992, 1995) and Bolduc, Fortin, and Gordon (1997). Today, however, eigenfunction-based spatial filtering offers a methodology that can account for spatial autocorrelation effects within a spatial interaction model. This paper updates work from the early 1980s, extending it with spatial filtering methods. Keywords: Distance decay; Gravity model; Spatial autocorrelation; Spatial filter; Spatial interaction JEL classification: C21; J20; R49
This paper was the source of a colloquium presentation to the Institute for Economic Geography and GIScience, Vienna University of Economics and Business Administration, on December 6, 2005. Funding for the project was obtained through the author’s position as the Ashbel Smith Professor.
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1. INTRODUCTION AND BACKGROUND The (unconstrained) gravity model of spatial interaction describes flows (Fij) between origin location i and destination location j with the following equation (see Sen and Smith 1995): (1)
Fij = κOiα D βj e
− γd ij
+ ε ij ,
where Oi denotes a mass term (e.g., population) for origin i, Dj denotes a mass term for destination j, dij denotes the distance separating locations i and j, εij denotes the random noise for the flow between locations i and j, and κ, α, β, and γ are parameters. Wilson (1970) establishes the exponential specification of the distance function. Meanwhile, the four parameters may be interpreted as follows. κ is a constant of proportionality that rescales the product of two quantities, α is an exponent that inflates a too-small Oi and deflates a too-large Oi, β is an exponent that inflates a too-small Di and deflates a too-large Di, and γ is a friction of distance measure that increases as distance becomes increasingly restrictive to the movement of flows. With only four parameters estimated from n(n – 1) flows—within areal unit Fii flows almost always are set aside—equation (1) furnishes a surprisingly good description of a disparate set of flows across a wide range of geographic landscapes. One weakness of equation (1) is that it overlooks spatial autocorrelation contained in its origin and its destination geographic distributions. Based upon the spatial linear operator commonly employed today in spatial statistics and spatial econometrics, Griffith and Jones (1980, p. 188) argue that, using matrix notation, a specification involving the following was needed: (2)
(I – ρoW)F and F(I - ρdW),
where I is an n-by-n identity matrix; F is the n-by-n flows matrix (with its diagonal coded with a missing value); W is a row-standardized geographic connectivity matrix (i.e., based upon a binary 0-1 connectivity matrix C for which cij = 1 if areal units i and j are neighbors, and 0 otherwise); and ρo and ρd, respectively, are the spatial autocorrelation parameters for the origin and the destination geographic distributions. Concerned about this problem, Griffith and Jones (1980) explore the relationship between spatial structure and spatial interaction at the intra-urban level by examining journey-to-work data for 24 selected Canadian cities. Their findings conclude that distance-decay exponents are
The Review of Regional Studies, Vol. 37, No. 1, 2007, pp. 28 – 38
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strongly influenced by geographic structure and the geometry of origins and destinations. They consistently uncover statistically significant, positive spatial autocorrelation in gravity model parameters. But computing power in the late 1970s and early 1980s was insufficient to estimate their proposed model specification, which involves an n2-by-n2 matrix. More recently, however, LeSage and Pace (2005) outline feasible methodology for estimating this revised equation. The purpose of this paper is to revisit the spatial-structure spatial-interaction problem by expanding the LeSage-Pace formulation to include a spatial filtering specification. Results are applied to German employment flows data. 2. SPATIAL FILTERING: AN OVERVIEW
One of the difficulties associated with spatial statistics revolves around its implementation. One of the attempts to address this problem has fostered the development of non- and semi-parametric spatial filtering techniques. These furnish an approach to dealing with spatial autocorrelation in regression analysis by separating a variable’s spatial and aspatial effects, allowing spatial statistical analysts to use conventional regression models for their data analyses. Four types of spatial filters exist for georeferenced data analysis, namely spatial linear operators (e.g., Griffith 1979), Getis’s Gi specification (e.g., Getis and Griffith 2002), Griffith’s eigenfunction specification, and the Legendregroup’s PCNM specification (e.g., Griffith and Peres-Neto 2006). Not surprisingly, all four of these versions are variants of the same basic conceptualization. The Griffith specification, which continues to be developed by Tiefelsdorf and Griffith (2007), is a transformation procedure that depends on mathematical expressions, known as eigenfunctions, that characterize the aforementioned geographic connectivity matrix C. This transformed connectivity matrix appears in the numerator of the Moran Coefficient (MC) spatial autocorrelation test statistic. This decomposition rests on the following property: when mapped spatially, the first eigenvector, say E1, is the set of real numbers that has the largest MC value achievable by any set of real numbers for the spatial arrangement defined by the geographic connectivity matrix C; the second eigenvector is the set of real numbers that has the largest achievable MC value by any set that is uncorrelated with E1; the third eigenvector is the third such set of values; and so on through En, the set of real numbers that has the largest negative MC value achievable by any set that is uncorrelated with the preceding (n – 1) eigenvectors. As such, these eigenvectors furnish distinct map pattern descriptions of latent spatial autocorrelation in georeferenced variables, because they are both orthogonal and uncorrelated. Their corresponding eigenvalues index the nature and degree of spatial autocorrelation portrayed by each eigenvector. The spatial filter is constructed from some linear combination of a subset of these eigenvectors. A stepwise regression procedure can be used to select those eigenvectors that account for the spatial autocorrelation in a response variable. This stepwise selection
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can be based upon the conventional residual sum-of-squares-minimization criterion or, as proposed by Tiefelsdorf and Griffith (2007), a residual MC-minimization criterion. 3. SPATIAL AUTOCORRELATION IN ORIGIN AND DESTINATION ATTRIBUTE VARIABLES
Daily German home-to-work trip commuting flows data for the single year of 2002 and the geographic resolution of NUTS-3 (the 439 German kreises) constitute the empirical basis of analyses summarized here (see Patuelli et al. 2006). These data were collected by the (German) Federal Employment Services (Bundesanstalt für Arbeit, BA). Each origin mass term was calculated by summing all entries in its corresponding row of the flows matrix. The geographic distributions of these mass terms—namely, the logarithms of the numbers of employees and of jobs—are portrayed in Figure 1. Each destination mass term was calculated by summing all entries in its corresponding column of the flows matrix. Once these origin and destination terms were computed, attention was restricted to the n(n – 1) = 192,282 off-diagonal values in the flows matrix. The total number of workers geographically distributed across the origins/destinations is 27,454,023; the total number of inter-kreise flows is 9,685,399. Separation was measured as the distance in thousands of kilometers between kreise centroids (computed in ArcGIS). Poisson regression estimation of equation (1) with and without the diagonal entries yields the following. Parameter
κˆ αˆ βˆ γˆ 2
Pseudo-R
With Diagonal Flows Without Diagonal Flows 0.0372 0.0001 0.4740 0.5262 0.7295 1.0010 3.4787 0.9188
1.6837 0.5067
Substantial overdispersion is present in these data (deviance statistic = 201 and 102, respectively) that cannot be accounted for by replacing the Poisson assumption with a negative binomial assumption. (The deviance statistic decreases to roughly 1, but the pseudo-R2, for example, plummets to approximately 0.5 and 0.2, respectively). This situation suggests the presence of spatial autocorrelation complications in these data that need to be addressed by accounting for this spatial autocorrelation. Parameter estimates for this unconstrained gravity model suggest that the size of labor pools for the origins are too large and need to be deflated (i.e., αˆ < 1), whereas the number of employment opportunities at the destinations are appropriate for the observed
inter-kreise journey-to-work flows (i.e., βˆ = 1). Not surprisingly, the distance decay parameter (i.e., - γˆ
